Abstract

A ratio-dependent predator-prey system with Holling type III functional response and feedback controls is proposed. By constructing a suitable Lyapunov function and using the comparison theorem of difference equation, sufficient conditions which ensure the permanence and global attractivity of the system are obtained. After that, under some suitable conditions, we show that the predator species 𝑦 will be driven to extinction. Examples together with their numerical simulations show that the main results are verifiable.

1. Introduction

Wang and Li [1, 2] established verifiable criteria for the existence of globally attractive positive periodic solutions of the following delayed predator-prey model with Holling type III functional response: 𝑁1(𝑡)=𝑁1𝑏(𝑡)1(𝑡)𝑎1(𝑡)𝑁1(𝑡𝜏1𝛼(𝑡))1(𝑡)𝑁1(𝑡)1+𝑚𝑁21𝑁(𝑡)2,𝑁(𝑡𝜎(𝑡))2(𝑡)=𝑁2(𝑡)𝑏2(𝑡)𝑎2(𝑡)𝑁2𝛼(𝑡)+2(𝑡)𝑁21(𝑡𝜏2(𝑡))1+𝑚𝑁21(𝑡𝜏2,(𝑡))(1.1) where 𝑁1(𝑡),𝑁2(𝑡) are the densities of the prey population and predator population at time 𝑡, respectively; 𝑏𝑖,𝑎𝑖,𝜏𝑖,𝜎,𝛼𝑖[0,+)(𝑖=1,2) are continuous functions of period 𝑇 and 𝑇0𝑏𝑖(𝑡)𝑑𝑡>0,𝛼𝑖(𝑡)0; 𝑚 is a nonnegative constant. For more works on the predator-prey system with Holling type functional response, one could refer to [311] and the references cited therein. But, recently, lots of scholars found that when predators have to search for food (and, therefore, have to share or compete for food), a more suitable general predator-prey theory should be based on the so-called ratio-dependent theory, which can be roughly stated as that the per capita predator growth rate should be the so-called ratio-dependent functional response. This is strongly supported by numerous fields and laboratory experiments and observations [12, 13]. In [14], Wang and Li proposed the following ratio-dependent predator-prey system with Holling type III functional response:𝑥(𝑡)=𝑥(𝑡)𝑎(𝑡)𝑏(𝑡)𝑡𝑘(𝑡𝑠)𝑥(𝑠)𝑑𝑠𝑐(𝑡)𝑥2(𝑡)𝑦(𝑡)𝑚2𝑦2(𝑡)+𝑥2,𝑦(𝑡)(𝑡)=𝑦(𝑡)𝑒(𝑡)𝑥2(𝑡𝜏)𝑚2𝑦2(𝑡𝜏)+𝑥2,(𝑡𝜏)𝑑(𝑡)(1.2)where 𝑥(𝑡),𝑦(𝑡) are the densities of the prey population and predator population at time 𝑡, respectively, 𝑎(𝑡),𝑏(𝑡),𝑐(𝑡),𝑑(𝑡), and 𝑒(𝑡) are all positive periodic continuous functions, and 𝑚>0,𝜏0 are real constants. They found that the criteria for the permanence are exactly the same as those for the existence of positive periodic solution of (1.2). For more works on the ratio-dependent predator-prey system, one could refer to [15, 16] and the references cited therein.

On the other hand, when the size of the population is rarely small or the population has nonoverlapping generations, the discrete time models are more appropriate than the continuous ones [17, 18]. For the discrete ratio-dependent predator-prey model with Holling type III functional response, Fan and Li [19] considered the following system: 𝑁1(𝑘+1)=𝑁1𝑏(𝑘)exp1(𝑘)𝑎1(𝑘)𝑁1(𝑘[𝜏1𝑎])1(𝑘)𝑁1(𝑘)𝑁2(𝑘)𝑁21(𝑘)+𝑚2(𝑘)𝑁22,𝑁(𝐾)2(𝑘+1)=𝑁2(𝑘)exp𝑏2𝑎(𝑘)+2(𝑘)𝑁21(𝑘[𝜏2])𝑁21(𝑘[𝜏1])+𝑚2(𝑘)𝑁22(𝑘[𝜏2,])(1.3) sufficient conditions which ensure the permanence of system (1.3) are obtained.

However, as pointed out by Huo and Li [20], “ecosystem in the real world is continuously disturbed by unpredictable forces which can result in changes in the biological parameters such as survival rates. The question of whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time is of practical interest in ecology . In the language of control variables, we call the disturbance functions as control variables.” For this direction, in [15], Chen and Ji proposed the following system:̇𝑥1=𝑥1𝑎1𝑐(𝑡)𝑏(𝑡)𝑥1(𝑡)𝑥1+𝛼(𝑡)𝑥2𝑒1(𝑡)𝑢1,̇𝑥2=𝑥2𝑎2𝑐(𝑡)+2(𝑡)𝑥1+𝛼(𝑡)𝑥2+𝑒2(𝑡)𝑢2,̇𝑢1=1(𝑡)𝑑1(𝑡)𝑢1+𝑓1(𝑡)𝑥1,̇𝑢2=2(𝑡)𝑑2(𝑡)𝑢2+𝑓2(𝑡)𝑥2,(1.4)where 𝑥𝑖(𝑡) stand for the densities of the prey and predator, respectively, 𝑢𝑖(𝑡)(𝑖=1,2) is the control variable, 𝛼(𝑡),𝑏(𝑡),𝑎𝑖(𝑡),𝑐𝑖(𝑡),𝑖(𝑡),𝑑𝑖(𝑡),𝑓𝑖(𝑡), and 𝑒𝑖(𝑡)(𝑖=1,2) are continuous and strictly positive functions. In this paper, they considered the almost periodic solution of system (1.4). For more work on this direction, one could refer to [15, 2128].

To the best of the author's knowledge, so far no scholar has considered system (1.3) with feedback controls. This motivates us to propose and study the following discrete ratio-dependent predator-prey system with Holling type III and feedback controls: 𝑥(𝑛+1)=𝑥(𝑛)exp𝑎(𝑛)𝑏(𝑛)𝑥(𝑛)𝑐(𝑛)𝑥(𝑛)𝑦(𝑛)𝑥2(𝑛)+𝑚2(𝑛)𝑦2(𝑛)𝑒1(𝑛)𝑢1,(𝑛)𝑦(𝑛+1)=𝑦(𝑛)exp𝑑(𝑛)+𝑓(𝑛)𝑥2(𝑛)𝑥2(𝑛)+𝑚2(𝑛)𝑦2(𝑛)𝑒2(𝑛)𝑢2,(𝑛)Δ𝑢1(𝑛)=𝜂1(𝑛)𝑢1(𝑛)+𝑞1(𝑛)𝑥(𝑛),Δ𝑢2(𝑛)=𝜂2(𝑛)𝑢2(𝑛)+𝑞2(𝑛)𝑦(𝑛),(1.5) where 𝑥(𝑡),𝑦(𝑡) are the densities of the prey population and predator population at time 𝑡, respectively, for 𝑖=1,2,{𝑎(𝑛)},{𝑏(𝑛)},{𝑐(𝑛)},{𝑑(𝑛)},{𝑓(𝑛)},{𝑚(𝑛)},{𝑒𝑖(𝑛)},{𝜂𝑖(𝑛)}, and {𝑞𝑖(𝑛)} are all bounded nonnegative sequences such that0<𝑎𝐿𝑎(𝑛)𝑎𝑈,0<𝑏𝐿𝑏(𝑛)𝑏𝑈,0<𝑐𝐿𝑐(𝑛)𝑐𝑈,0<𝑑𝐿𝑑(𝑛)𝑑𝑈,0<𝑓𝐿𝑓(𝑛)𝑓𝑈,0<𝑚𝐿𝑚(𝑛)𝑚𝑈,0<𝑒𝐿𝑖𝑒𝑖(𝑛)𝑒𝑈𝑖,0<𝜂𝐿𝑖𝜂𝑖(𝑛)𝜂𝑈𝑖<1,0<𝑞𝐿𝑖𝑞𝑖(𝑛)𝑞𝑈𝑖(<1.H0)Here, for any bounded sequence {𝑎(𝑛)},𝑎𝐿=inf𝑛{𝑎(𝑛)},𝑎𝑈=sup𝑛{𝑎(𝑛)}.

By the biological meaning, we will focus our discussion on the positive solutions of system (1.5). So, it is assumed that the initial conditions of system (1.5) are of the form𝑥(0)>0,𝑦(0)>0,𝑢𝑖(0)>0,𝑖=1,2.(1.6)It is not difficult to see that solutions of (1.5) and (1.6) are well defined and satisfy 𝑥(𝑛)>0,𝑦(𝑛)>0,𝑢𝑖(𝑛)>0,for𝑘+.(1.7)

The main purpose of this paper is to derive sufficient conditions for the permanence, global attractivity, and extinction of system (1.5).

The organization of this paper is as follows. In Section 2, we introduce some useful lemmas. In Section 3, we will study the permanence and global attractivity of system (1.5). In Section 4, we will study the extinction of the predator species 𝑦. In the last section, numerical simulation is presented to illustrate the feasibility of our main results.

2. Preliminaries

Now, let us state several lemmas which will be useful in proving the main results.

First, let us consider the first-order difference equation𝑦(𝑘+1)=𝐴𝑦(𝑘)+𝐵,𝑘=1,2,(2.1)where 𝐴,𝐵 are positive constants. The following Lemma 2.1 is a direct corollary of Theorem 6.2 of L. Wang and M. Q. Wang [29, page 125].

Lemma 2.1. Assume that |𝐴|<1. For any initial 𝑦(0), there exists a unique solution 𝑦(𝑘) of (2.1), which can be expressed as follows: 𝑦(𝑘)=𝐴𝑘𝑦(0)𝑦+𝑦,(2.2)where 𝑦=𝐵/(1𝐴). Thus, for any solutions 𝑦(𝑘) of system (2.1), lim𝑘+𝑦(𝑘)=𝑦.(2.3)

The following comparison theorem for difference equation is Theorem 2.1 of [29, page 241].

Lemma 2.2. Let 𝑘+𝑘0={𝑘0,𝑘0+1,,𝑘0+𝑙,} and 𝑟0. For any fixed 𝑘,𝑔(𝑘,𝑟) is a nondecreasing function with respect to 𝑟, and for 𝑘𝑘0, the following inequalities hold: 𝑦(𝑘+1)𝑔(𝑘,𝑦(𝑘)),𝑢(𝑘+1)𝑔(𝑘,𝑢(𝑘)).(2.4)If 𝑦(𝑘0)𝑢(𝑘0), then 𝑦(𝑘)𝑢(𝑘) for all 𝑘𝑘0.

The following Lemmas 2.3 and 2.4 can be found in [21].

Lemma 2.3 (see [21]). Assume that 𝑥(𝑛) satisfies 𝑥(𝑛)>0 and 𝑥(𝑛+1)𝑥(𝑛)exp(𝑟(𝑛)𝑎(𝑛)𝑥(𝑛))(2.5)for 𝑛, where 𝑟(𝑛) and 𝑎(𝑛) are nonnegative sequences bounded above and below by positive constants. Then limsup𝑛+1𝑥(𝑛)𝑎𝐿exp(𝑟𝑈1).(2.6)

Lemma 2.4 (see [21]). Assume that {𝑥(𝑛)} satisfies 𝑥(𝑛+1)𝑥(𝑛)exp(𝑟(𝑛)𝑎(𝑛)𝑥(𝑛)),𝑘𝑁0,(2.7)limsup𝑛+𝑥(𝑛)𝑥 and 𝑥(𝑁0)>0, where 𝑟(𝑛) and 𝑎(𝑛) are nonnegative sequences bounded above and below by positive constants and 𝑁0. Then liminf𝑛+𝑟𝑥(𝑛)𝐿𝑟exp𝐿𝑎𝑈𝑥𝑎𝑈.(2.8)

3. Permanence and Global Attractivity

Now, we investigate the permanence and global attractivity of system (1.5).

Theorem 3.1. Assume that 𝑎𝐿𝑐𝑈2𝑚𝐿𝑒𝑈1𝑊1>0,(H1)𝑓𝐿𝑑𝑈𝑒𝑈2𝑊2>0(H2)𝑥hold, then the species 𝑦 and (𝑥(𝑛),𝑦(𝑛),𝑢1(𝑛),𝑢2(𝑛)) are permanent, that is, for any positive solution 𝑚1liminf𝑛+𝑥(𝑛)limsup𝑛+𝑥(𝑛)𝑀1,𝑚2liminf𝑛+𝑦(𝑛)limsup𝑛+𝑦(𝑛)𝑀2,𝑤1liminf𝑛+𝑢1(𝑛)limsup𝑛+𝑢1(𝑛)𝑊1,𝑤2liminf𝑛+𝑢2(𝑛)limsup𝑛+𝑢2(𝑛)𝑊2,(3.1) of system (1.5) with the initial conditions (1.6), 𝑚1=𝑎𝐿𝑐𝑈/2𝑚𝐿𝑒𝑈1𝑊1𝑏𝑈𝑎exp𝐿𝑏𝑈𝑀1𝑐𝑈2𝑚𝐿𝑒𝑈1𝑊1,𝑚2=2𝑚1(𝑓𝐿𝑑𝑈𝑒𝑈2𝑊2)𝑓𝐿𝑚𝑈exp𝑑𝑈𝑒2𝑊2,𝑤1=𝑞𝐿1𝑚1𝜂𝑈1,𝑤2=𝑞𝐿2𝑚2𝜂𝑈2,𝑀1=1𝑏𝐿𝑎exp𝑈1,𝑀2=𝑓𝑈𝑀12𝑑𝐿𝑚𝐿𝑓exp𝑈𝑑𝐿,𝑊1=𝑞𝑈1𝑀1𝜂𝐿1,𝑊2=𝑞𝑈2𝑀2𝜂𝐿2.(3.2)where limsup𝑛+𝑥(𝑛)𝑀1.(3.3)

Proof. We divided the proof into five steps.
Step 1. We show𝑥(𝑛+1)𝑥(𝑛)exp𝑎(𝑛)𝑏(𝑛)𝑥(𝑛).(3.4) From the first equation of system (1.5),𝜀Then (3.3) follows immediately from Lemma 2.3. Thus, for any positive 𝑛0>0, there exists an 𝑥(𝑛)𝑀1+𝜀𝑛>𝑛0.(3.5) such thatlimsup𝑛+𝑦(𝑛)𝑀2Step 2. We prove 𝑛1𝑛0 by distinguishing two cases.
Case 1. There exists an 𝑦(𝑛1+1)𝑦(𝑛1) such that 𝑑(𝑛1)+𝑓(𝑛1)𝑥2(𝑛1)𝑥2(𝑛1)+𝑚2(𝑛1)𝑦2(𝑛1)𝑒2(𝑛1)𝑢2(𝑛1)0,(3.6). Then, by the second equation of system (1.5), we have𝑑𝐿+𝑓𝑈𝑥(𝑛1)2𝑚𝐿𝑦(𝑛1)0.(3.7)which implies𝑦(𝑛1)𝑓𝑈(𝑀1+𝜀)/2𝑚𝐿𝑑𝐿The above inequality combined with (3.5) leads to 𝑦(𝑛1+1)𝑦(𝑛1)exp𝑑(𝑛1)+𝑓(𝑛1)𝑥2(𝑛1)𝑥2(𝑛1)+𝑦2(𝑛1)𝑦(𝑛1)exp𝑑𝐿+𝑓𝑈𝑓𝑈(𝑀1+𝜀)2𝑚𝐿𝑑𝐿exp𝑑𝐿+𝑓𝑈def=𝑀𝜀2.(3.8). Thus from the second equation of system (1.5) again we have𝑦(𝑛)𝑀𝜀2We claim that 𝑛𝑛1 for all 𝑞𝑛1+2. In fact, suppose there exists 𝑦(𝑞)>𝑀𝜀2 such that 𝑞0. Let 𝑛1 be the smallest integer between 𝑞 and 𝑦(𝑞0)>𝑀𝜀2 such that 𝑦(𝑞01)𝑀𝜀2 and 𝑦(𝑞0)>𝑦(𝑞01). Then 𝑦(𝑞0)𝑀𝜀2 implies 𝜀0, a contradiction. This proves the claim. Setting limsup𝑛+𝑦(𝑛)𝑀2 in (3.8) leads to 𝑦(𝑛+1)𝑦(𝑛).Case 2. Suppose 𝑛𝑛0 for all lim𝑛+𝑦(𝑛). In this case, 𝑦 exists, denoted by 𝑦𝑓𝑈𝑀1/2𝑚𝐿𝑑𝐿. We claim that 𝑦>𝑓𝑈𝑀1/2𝑚𝐿𝑑𝐿. If not, suppose 𝜎>0. Choose 𝜎<𝑦𝑓𝑈𝑀1/2𝑚𝐿𝑑𝐿 such that lim𝑛+𝑑(𝑛)+𝑓(𝑛)𝑥2(𝑛)𝑥2(𝑛)+𝑚2(𝑛)𝑦2(𝑛)𝑒2(𝑛)𝑢2(𝑛)=0,(3.9). Taking limit in the second equation of system (1.5) produces𝑑(𝑛)+𝑓(𝑛)𝑥2(𝑛)𝑥2(𝑛)+𝑚2(𝑛)𝑦2(𝑛)𝑒2(𝑛)𝑢2(𝑛)𝑑𝐿+𝑓𝑈𝑥(𝑛)2𝑚𝐿𝑦(𝑛)<𝑑𝐿+𝑓𝑈𝑀12𝑚𝐿(𝑦𝜎)<0(3.10)which is impossible as𝑛for sufficiently large 𝑓𝑈𝑀1/2𝑚𝐿𝑑𝐿𝑀2. This proves the claim. Since (H2) implies limsup𝑛+𝑦(𝑛)𝑀2, we have proved limsup𝑛+𝑢1(𝑛)𝑊1,limsup𝑛+𝑢2(𝑛)𝑊2.(3.11).
Step 3. We verify𝜀For any positive 𝑛2, there exists 𝑥(𝑛)𝑀1+𝜀,𝑦(𝑛)𝑀2+𝜀for𝑛𝑛2.(3.12) such that𝑛𝑛2For Δ𝑢1(𝑛)𝜂1(𝑛)𝑢1(𝑛)+𝑞1(𝑛)(𝑀1+𝜀).(3.13), (3.12) combined with the third equation of (1.5) gives𝑢1(𝑛+1)1𝜂𝐿1𝑢1(𝑛)+𝑞𝑈1(𝑀1+𝜀)for𝑛𝑛2.(3.14)Thuslimsup𝑛+𝑢1𝑞(𝑛)𝑈1(𝑀1+𝜀)𝜂𝐿1.(3.15)With the help of Lemmas 2.1 and 2.2, we obtain𝜀0Letting limsup𝑛+𝑢1(𝑛)𝑊1.(3.16), we immediately getlimsup𝑛+𝑢2(𝑛)𝑊2Similarly, one can show liminf𝑛+𝑥(𝑛)𝑚1,limsup𝑛+𝑦(𝑛)𝑚2.(3.17).Step 4. We check𝑎𝐿𝑐𝑈2𝑚𝐿𝑒𝑈1(𝑊1+𝜀)>0,𝑓𝐿𝑑𝑈𝑒𝑈2(𝑊1+𝜀)>0(3.18)Conditions (H1) and (H2) imply that𝜀hold for all small enough positive constant 𝜀. For any such 𝑛3, there exists 𝑥(𝑛)𝑀1+𝜀,𝑢1(𝑛)𝑊1+𝜀for𝑛𝑛3.(3.19) such that𝑛𝑛3Then, for 𝑎𝑥(𝑛+1)𝑥(𝑛)exp𝐿𝑐𝑈2𝑚𝐿𝑒𝑈1(𝑊1+𝜀)𝑏𝑈𝑥(𝑛).(3.20), it follows from (3.19) and the first equation of system (1.5) thatliminf𝑛+𝑎𝑥(𝑛)𝐿𝑐𝑈/2𝑚𝐿𝑒𝑈1(𝑊1+𝜀)𝑏𝑈𝑎exp𝐿𝑐𝑈2𝑚𝐿𝑒𝑈1(𝑊1+𝜀)𝑏𝑈(𝑀1+𝜀).(3.21)According to Lemma 2.4, one has𝜀0Letting liminf𝑛+𝑥(𝑛)𝑚1.(3.22) leads to𝜀1<𝑚1/2
Now, for any small positive 𝑛4>0, it follows from (3.19) and (3.22) that there exists 𝑥(𝑛)𝑚1𝜀1,𝑦(𝑛)𝑀2+𝜀1,𝑢2(𝑛)𝑊2+𝜀1(3.23) such that𝑛𝑛4for 𝑦(𝑛+1)𝑦(𝑛)exp𝑑𝑈+𝑓𝐿𝑒𝑈2(𝑊2+𝜀1𝑓)𝐿𝑚𝑈𝑦(𝑛)2(𝑚1𝜀1).(3.24). This, combined with the second equation of system (1.5), givesliminf𝑛+𝑦(𝑛)2(𝑚1𝜀1)(𝑓𝐿𝑑𝑈𝑒𝑈2(𝑊2+𝜀1))𝑓𝐿𝑚𝑈𝑓exp𝐿𝑑𝑈𝑒𝑈2(𝑊2+𝜀1𝑓)𝐿𝑚𝑈(𝑀2+𝜀1)2(𝑚1𝜀1).(3.25)Applying Lemma 2.4, one easily obtains𝜀Because of the arbitrariness of liminf𝑛+𝑦(𝑛)𝑚2, it is not difficult to see that liminf𝑛+𝑢1(𝑛)𝑤1.
Step 5. Finally, we only show liminf𝑛+𝑢2(𝑛)𝑤2 as the proof of 𝜀2>0 is similar. For any 𝜀2<𝑚1 such that 𝑛5, there exists 𝑥(𝑛)𝑚1𝜀2for𝑛𝑛5.(3.26) such thatΔ𝑢1(𝑛)𝜂1(𝑛)𝑢1(𝑛)+𝑞𝐿1(𝑚1𝜀2)(3.27)This and the third equation of system (1.5) imply that𝑛𝑛5for 𝑛𝑛5. Then, for 𝑢1(𝑛+1)(1𝜂𝑈1)𝑢1(𝑛)+𝑞𝐿1(𝑚1𝜀2).(3.28),liminf𝑛+𝑢1𝑞(𝑛)𝐿1(𝑚1𝜀2)𝜂𝑈1.(3.29)It follows from Lemmas 2.1 and 2.2 immediately that𝜀20Letting liminf𝑛+𝑢1(𝑛)𝑤1 gives 𝛼,𝛽,𝛾,𝜁. This completes the proof of the theorem.

Theorem 3.2. Assume that (H1) and (H2) hold. Assume further that there exist positive constants 𝛿, and 𝑏𝛼min𝐿,2𝑀1𝑏𝑈𝑐𝛼𝑈4(𝑚𝐿)2𝑚2𝑐𝛼𝑈𝑀14𝑚12𝑓𝛽𝑈𝑀22𝑚2𝑚1𝛾𝑞𝑈1>𝛿,(H3)𝛽min2𝑓𝐿(𝑚𝐿)2𝑚21𝑚2𝑀21+(𝑚𝑈)2𝑀222,2𝑀2𝑓𝑈𝑀12𝑚1𝑚2𝑐𝛼𝑈𝑀14(𝑚𝐿)2𝑚22𝑐𝛼𝑈4𝑚1𝜁𝑞𝑈2(>𝛿,H4)𝛾𝜂𝐿1𝛼𝑒𝑈1(>𝛿,H5)𝜁𝜂𝐿2𝛽𝑒𝑈2(>𝛿.H6) such that 𝑥𝑦(𝑥1(𝑛),𝑦1(𝑛),𝑢1(𝑛),𝑢2(𝑛))(𝑥2(𝑛),𝑦2(𝑛),𝑢1(𝑛),𝑢2(𝑛))Then the species lim𝑛+|||𝑥1(𝑛)𝑥2|||(𝑛)=0,lim𝑛+|||𝑦1(𝑛)𝑦2|||(𝑛)=0,lim𝑛+|||𝑢1(𝑛)𝑢1|||(𝑛)=0,lim𝑛+|||𝑢2(𝑛)𝑢2|||(𝑛)=0.(3.30) and 𝜀<min{𝑚1/2,𝑚2/2} are globally attractive, that is, for any positive solutions 𝑏𝛼min𝐿,2(𝑀1+𝜀)𝑏𝑈𝑐𝛼𝑈4(𝑚𝐿)2(𝑚2𝑐𝜀)𝛼𝑈(𝑀1+𝜀)4(𝑚1𝜀)2𝑓𝛽𝑈(𝑀2+𝜀)2(𝑚1𝜀)(𝑚2𝜀)𝛾𝑞𝑈1>𝛿,𝛽min2𝑓𝐿(𝑚𝐿)2(𝑚1𝜀)2(𝑚2𝜀)((𝑀1+𝜀)2+(𝑚𝑈)2(𝑀2+𝜀)2)2,2(𝑀2𝑓+𝜀)𝑈(𝑀1+𝜀)2(𝑚1𝜀)(𝑚2𝑐𝜀)𝛼𝑈(𝑀1+𝜀)4(𝑚𝐿)2(𝑚2𝜀)2𝑐𝛼𝑈4(𝑚1𝜀)𝜁𝑞𝑈2>𝛿,𝛾𝜂𝐿1𝛼𝑒𝑈1>𝛿,𝜁𝜂𝐿2𝛽𝑒𝑈2>𝛿.(3.31) and (𝑥1(𝑛),𝑦1(𝑛),𝑢1(𝑛),𝑢2(𝑛)) of system (1.5) with the initial conditions (1.6), (𝑥2(𝑛),𝑦2(𝑛),𝑢1(𝑛),𝑢2(𝑛))

Proof. From conditions (H3)–(H6), there exists small enough positive constant 𝑚1liminf𝑛+𝑥𝑖(𝑛)limsup𝑛+𝑥𝑖(𝑛)𝑀1,𝑚2liminf𝑛+𝑦𝑖(𝑛)limsup𝑛+𝑦𝑖(𝑛)𝑀2,𝑤𝑖liminf𝑛+𝑢𝑖(𝑛)limsup𝑛+𝑢𝑖(𝑛)𝑊𝑖,𝑤𝑖liminf𝑛+𝑢𝑖(𝑛)limsup𝑛+𝑢𝑖(𝑛)𝑊𝑖,𝑖=1,2.(3.32) such that𝑛0>0Since (H1) and (H2) hold, for any positive solutions 𝑛>𝑛0 and 𝑚1𝜀𝑥𝑖(𝑛)𝑀1+𝜀,𝑚2𝜀𝑦𝑖(𝑛)𝑀2𝑤+𝜀,𝑖𝜀𝑢𝑖(𝑛)𝑊𝑖+𝜀,𝑤𝑖𝜀𝑢𝑖(𝑛)𝑊𝑖+𝜀,𝑖=1,2.(3.33) of system (1.5) with the initial conditions (1.6), it follows from Theorem 3.1 that𝑉1||(𝑛)=ln𝑥1(𝑛)ln𝑥2||(𝑛).(3.34)Then, there exists an 𝑉1||(𝑛+1)=ln𝑥1(𝑛+1)ln𝑥2||||(𝑛+1)ln𝑥1(𝑛)ln𝑥2(𝑛)𝑏(𝑛)(𝑥1(𝑛)𝑥2|||||𝑥(𝑛))+𝑐(𝑛)1(𝑛)𝑦1(𝑛)𝑥21(𝑛)+𝑚2(𝑛)𝑦21𝑥(𝑛)2(𝑛)𝑦2(𝑛)𝑥22(𝑛)+𝑚2(𝑛)𝑦22|||(𝑛)+𝑒1||𝑢(𝑛)1(𝑛)𝑢1||.(𝑛)(3.35) such that, for all 𝑥1(𝑛)𝑥2(𝑛)=exp(ln𝑥1(𝑛))exp(ln𝑥2(𝑛))=𝜉1(𝑛)(ln𝑥1(𝑛)ln𝑥2(𝑛)),(3.36),𝜉1(𝑛)Let𝑥1(𝑛)Then, from the first equation of system (1.5), we have𝑥2(𝑛)Using the mean value theorem, we get𝑉1||(𝑛+1)ln𝑥1(𝑛)ln𝑥2||1(𝑛)𝜉1|||1(𝑛)𝜉1|||(𝑛)𝑏(𝑛)|𝑥1(𝑛)𝑥2||+|||(𝑛)𝑐(𝑛)𝑥1(𝑛)𝑦1(𝑛)𝑥2(𝑛)(𝑥21(𝑛)+𝑚2(𝑛)𝑦21(𝑛))(𝑥22(𝑛)+𝑚2(𝑛)𝑦22|||||𝑥(𝑛))1(𝑛)𝑥2||+|||(𝑛))𝑐(𝑛)𝑚2(𝑛)𝑦21(𝑛)𝑦2(𝑛)(𝑥21(𝑛)+𝑚2(𝑛)𝑦21(𝑛))(𝑥22(𝑛)+𝑚2(𝑛)𝑦22|||||𝑥(𝑛))1(𝑛)𝑥2||+|||(𝑛))𝑐(𝑛)𝑥21(𝑛)𝑥2(𝑛)(𝑥21(𝑛)+𝑚2(𝑛)𝑦21(𝑛))(𝑥22(𝑛)+𝑚2(𝑛)𝑦22|||||𝑦(𝑛))1(𝑛)𝑦2||+|||(𝑛))𝑐(𝑛)𝑚2(𝑛)𝑦1(𝑛)𝑦2(𝑛)𝑥1(𝑛)(𝑥21(𝑛)+𝑚2(𝑛)𝑦21(𝑛))(𝑥22(𝑛)+𝑚2(𝑛)𝑦22|||||𝑦(𝑛))1(𝑛)𝑦2||(𝑛))+𝑒1||𝑢(𝑛)1(𝑛)𝑢1||.(𝑛)(3.37)where 𝑛𝑛0 lies between Δ𝑉1𝑏(𝑛)min𝐿,2𝑀1+𝜀𝑏𝑈||𝑥1(𝑛)𝑥2||+𝑐(𝑛)𝑈4(𝑚𝐿)2(𝑚2||𝑥𝜀)1(𝑛)𝑥2||+𝑐(𝑛)𝑈(𝑀1+𝜀)4(𝑚1𝜀)2||𝑥1(𝑛)𝑥2||+𝑐(𝑛)𝑈(𝑀1+𝜀)4(𝑚𝐿)2(𝑚2𝜀)2||𝑦1(𝑛)𝑦2||+𝑐(𝑛)𝑈4(𝑚1||𝑦𝜀)1(𝑛)𝑦2||(𝑛)+𝑒𝑈1||𝑢1(𝑛)𝑢1||.(𝑛)(3.38) and 𝑉2||(𝑛)=ln𝑦1(𝑛)ln𝑦2||(𝑛).(3.39).
It follows from (3.35) and (3.36) that𝑉2=||(𝑛+1)ln𝑦1(𝑛+1)ln𝑦2||=|||(𝑛+1)ln𝑦1(𝑛)ln𝑦2𝑥(𝑛)+𝑓(𝑛)21(𝑛)𝑥21+𝑚2(𝑛)𝑦21𝑥(𝑛)22(𝑛)𝑥22(𝑛)+𝑚2(𝑛)𝑦22(𝑛)𝑒2(𝑛)(𝑢2(𝑛)𝑢2||||||(𝑛))ln𝑦1(𝑛)ln𝑦2(𝑛)𝑓(𝑛)𝑚2(𝑛)𝑥1(𝑛)(𝑥1(𝑛)𝑦2(𝑛)+𝑦1(𝑛)𝑥2(𝑛))(𝑥21(𝑛)+𝑚2(𝑛)𝑦21(𝑛))(𝑥22(𝑛)+𝑚2(𝑛)𝑦22(𝑛))×(𝑦1(𝑛)𝑦2|||+(𝑛))𝑓(𝑛)𝑚2(𝑛)𝑦1(𝑛)(𝑥1(𝑛)𝑦2(𝑛)+𝑦1(𝑛)𝑥2(𝑛))(𝑥21(𝑛)+𝑚2(𝑛)𝑦21(𝑛))(𝑥22(𝑛)+𝑚2(𝑛)𝑦22×||𝑥(𝑛))1(𝑛)𝑥2||(𝑛)+𝑒2||𝑢(𝑛)1(𝑛)𝑢1||.(𝑛)(3.40)So, for 𝑦1(𝑛)𝑦2(𝑛)=exp(ln𝑦1(𝑛))exp(ln𝑦2(𝑛))=𝜉2(𝑛)(ln𝑦1(𝑛)ln𝑦2(𝑛)),(3.41),𝜉2(𝑛)Let𝑦1(𝑛)Then, from the second equation of system (1.5), we have𝑦2(𝑛)Using the mean value theorem, we get𝑛>𝑛0where Δ𝑉21(𝑛)𝜉2|||1(𝑛)𝜉2(𝑛)𝑓(𝑛)𝑚2(𝑛)𝑥1(𝑛)(𝑥1(𝑛)𝑦2(𝑛)+𝑦1(𝑛)𝑥2(𝑛))(𝑥21(𝑛)+𝑚2(𝑛)𝑦21(𝑛))(𝑥22(𝑛)+𝑚2(𝑛)𝑦22|||||𝑦(𝑛))1(𝑛)𝑦2||+(𝑛)𝑓(𝑛)𝑚2(𝑛)𝑦1(𝑛)(𝑥1(𝑛)𝑦2(𝑛)+𝑦1(𝑛)𝑥2(𝑛))(𝑥21(𝑛)+𝑚2(𝑛)𝑦21(𝑛))(𝑥22(𝑛)+𝑚2(𝑛)𝑦22||𝑥(𝑛))1(𝑛)𝑥2||(𝑛)+𝑒2||𝑢(𝑛)2(𝑛)𝑢2||(𝑛)min2𝑓𝐿(𝑚𝐿)2(𝑚1𝜀)2(𝑚2𝜀)((𝑀1+𝜀)2+(𝑚𝑈)2(𝑀2+𝜀))2,2(𝑀2𝑓+𝜀)𝑈(𝑀1+𝜀)2(𝑚1𝜀)(𝑚2×||𝑦𝜀)1(𝑛)𝑦2||+𝑓(𝑛)𝑈(𝑀2+𝜀)2(𝑚1𝜀)(𝑚2||𝑥𝜀)1(𝑛)𝑥2||(𝑛)+𝑒𝑈2||𝑢2(𝑛)𝑢2||.(𝑛)(3.42) lies between 𝑉3|||𝑢(𝑛)=1(𝑛)𝑢1|||(𝑛).(3.43) and 𝑉3|||𝑢(𝑛+1)=1(𝑛+1)𝑢1|||(𝑛+1)(1𝜂1|||𝑢(𝑛))1(𝑛)𝑢1|||(𝑛)+𝑞1||𝑥(𝑛)1(𝑛)𝑥2||.(𝑛)(3.44).
Now, it follows from (3.40) and (3.41) that, for 𝑛>𝑛0,Δ𝑉3(𝑛)𝜂𝐿1|||𝑢1(𝑛)𝑢1|||(𝑛)+𝑞𝑈1|||𝑥1(𝑛)𝑥2|||(𝑛).(3.45)Let𝑉4|||𝑢(𝑛)=2(𝑛)𝑢2|||(𝑛).(3.46) Then, from the third equation of system (1.5), one can easily obtain thatΔ𝑉4(𝑛)𝜂𝐿2|||𝑢2(𝑛)𝑢2|||(𝑛)+𝑞𝑈2|||𝑦1(𝑛)𝑦2|||(𝑛).(3.47)It follows from (3.44) that, for 𝑉(𝑛)=𝛼𝑉1(𝑛)+𝛽𝑉2(𝑛)+𝛾𝑉3(𝑛)+𝜁𝑉4(𝑛).(3.48),𝑛>𝑛0Let𝑏Δ𝑉(𝑛)𝛼min𝐿,2(𝑀1+𝜀)𝑏𝑈𝑐𝛼𝑈4(𝑚𝐿)2(𝑚2𝑐𝜀)𝛼𝑈(𝑀1+𝜀)4(𝑚1𝜀)2𝑓𝛽𝑈(𝑀2+𝜀)2(𝑚1𝜀)(𝑚2𝜀)𝛾𝑞𝑈1×||𝑥1(𝑛)𝑥2||(𝑛)𝛽min2𝑓𝐿(𝑚𝐿)2(𝑚1𝜀)2(𝑚2𝜀)((𝑀1+𝜀)2+(𝑚𝑈)2(𝑀2+𝜀)2)2,2(𝑀2+𝜀)2𝑓𝑈(𝑀1+𝜀)2(𝑚1𝜀)(𝑚2𝑐𝜀)𝛼𝑈(𝑀1+𝜀)4(𝑚𝐿)2(𝑚2𝜀)2𝑐𝛼𝑈4(𝑚1𝜀)𝜁𝑞𝑈2×||𝑦1(𝑛)𝑦2||(𝑛)(𝛾𝜂𝐿1𝛼𝑒𝑢1|||𝑢)×1(𝑛)𝑢1|||(𝑛)(𝜁𝜂𝐿2𝛽𝑒𝑈2|||𝑢)×2(𝑛)𝑢2|||||𝑥(𝑛)𝛿(1(𝑛)𝑥2||+||𝑦(𝑛)1(𝑛)𝑦2||+|||𝑢(𝑛)1(𝑛)𝑢1|||+||𝑢(𝑛)2(𝑛)𝑢2||(𝑛)).(3.49) Similar to the analysis of (3.44) and (3.45), we can get𝑛0Now, we define a Lyapunov function as follows:𝑛For 𝑛𝑝=𝑛0(𝑉(𝑝+1)𝑉(𝑝))𝛿𝑛𝑝=𝑛0||𝑥1(𝑝)𝑥2||+||𝑦(𝑝)1(𝑝)𝑦2||+|||𝑢(𝑝)1(𝑝)𝑢1|||+|||𝑢(𝑝)2(𝑝)𝑢2|||,(𝑝)(3.50), it follows from (3.38), (3.42), (3.45), and (3.47) that𝑉(𝑛0)𝛿𝑛𝑝=𝑛0||𝑥1(𝑝)𝑥2||+||𝑦(𝑝)1(𝑝)𝑦2||+|||𝑢(𝑝)1(𝑝)𝑢1|||+|||𝑢(𝑝)2(𝑝)𝑢2|||(𝑝)+𝑉(𝑛+1).(3.51)Summating both sides of the above inequalities from 𝑉(𝑛0)𝛿𝑛𝑝=𝑛0||𝑥1(𝑝)𝑥2||+||𝑦(𝑝)1(𝑝)𝑦2||+|||𝑢(𝑝)1(𝑝)𝑢1|||+|||𝑢(𝑝)2(𝑝)𝑢2|||.(𝑝)(3.52) to 𝑛, we have𝑉(𝑛0)𝛿+𝑝=𝑛0|||𝑥1(𝑝)𝑥2|||+|||𝑦(𝑝)1(𝑝)𝑦2|||+|||𝑢(𝑝)1(𝑝)𝑢1|||+|||𝑢(𝑝)2(𝑝)𝑢2|||,(𝑝)(3.53)which implieslim𝑛|||𝑥1(𝑛)𝑥2|||+|||𝑦(𝑛)1(𝑛)𝑦2|||+|||𝑢(𝑛)1(𝑛)𝑢1|||+|||𝑢(𝑛)2(𝑛)𝑢2|||(𝑛)=0,(3.54)It follows thatlim𝑛|||𝑥1(𝑛)𝑥2|||(𝑛)=0,lim𝑛|||𝑦1(𝑛)𝑦2|||(𝑛)=0,lim𝑛|||𝑢1(𝑛)𝑢1|||(𝑛)=0,lim𝑛|||𝑢1(𝑛)𝑢1|||(𝑛)=0.(3.55)Letting 𝑦 gives𝑑𝐿+𝑓𝑈(<0.H7)which implies that(𝑥(𝑛),𝑦(𝑛),𝑢1(𝑛),𝑢2(𝑛))that is,lim𝑛+𝑦(𝑛)=0.(4.1)This completes the proof of Theorem 3.2.

4. Extinction of the Predator Species

This section is devoted to studying the extinction of the predator species 𝛾>0.

Theorem 4.1. Assume that 𝑑𝐿+𝑓𝑈<𝛾<0.(4.2)Then, for any solution 𝑛 of system (1.5), 𝑦(𝑛+1)=𝑦(𝑛)exp𝑑(𝑛)+𝑓(𝑛)𝑥2(𝑛)𝑥2(𝑛)+𝑚2(𝑛)𝑦2(𝑛)𝑒2(𝑛)𝑢2(𝑛)<𝑦(𝑛)exp𝑑𝐿+𝑓𝑈<𝑦(𝑛)exp𝛾}.(4.3)

Proof. From condition (H7), there exists small enough positive constant 𝑦(𝑛+1)<𝑦(0)exp𝑛𝛾,(4.4) such thatlim𝑛+𝑦(𝑛)=0.(4.5) For all 𝑥(𝑛+1)=𝑥(𝑛)exp1.225+0.025sin2𝑛(5.25+0.25cos(𝑛))𝑥(𝑛)((0.0075+0.0025sin(𝑛))𝑦(𝑛)𝑥(𝑛)𝑥2(𝑛)+0.885+0.005cos3𝑛2𝑦2(𝑛)0.002+0.001cos𝑢3𝑛1,(𝑛)𝑦(𝑛+1)=𝑦(𝑛)exp0.1125+0.0025cos+2𝑛0.195+0.005sin𝑥3𝑛2(𝑛)𝑥2(𝑛)+0.885+0.005cos3𝑛2𝑦2(𝑛)(0.0015+0.0005sin(𝑛))𝑢2,(𝑛)Δ𝑢1(𝑛)=(0.925+0.025sin(𝑛))𝑢1(𝑛)+(0.0375+0.0275cos(𝑛))𝑥(𝑛),Δ𝑢2(𝑛)=(0.925+0.025cos(𝑛))𝑢2(𝑛)+(0.025+0.005sin(𝑛))𝑦(𝑛).(5.1), from (4.2) and the second equation of system (1.5), one can easily obtain that𝛼=0.06,𝛽=0.05,𝛾=0.01,𝜁=0.001Therefore,𝛿=0.00001which yields𝑎𝐿𝑐𝑈2𝑚𝐿𝑒𝑈1𝑊1𝑓1.2>0,𝐿𝑑𝑈𝑒𝑈2𝑊2𝑏0.04>0,𝛼min𝐿,2𝑀1𝑏𝑈𝑐𝛼𝑈4(𝑚𝐿)2𝑚2𝑐𝛼𝑈𝑀14𝑚21𝑓𝛽𝑈𝑀22𝑚1𝑚2𝛾𝑞𝑈10.04512>𝛿,𝛽min2𝑓𝐿(𝑚𝐿)2𝑚21𝑚2𝑀21+(𝑚𝑈)2𝑀22,2𝑀2𝑓𝑈𝑀12𝑚1𝑚2𝑐𝛼𝑈𝑀14(𝑚𝐿)2𝑚2𝑐𝛼𝑈4𝑚1𝜁𝑞𝑈20.000013>𝛿,𝛾𝜂𝐿1𝛼𝑒𝑈10.0072>𝛿,𝜁𝜂𝐿2𝛽𝑒𝑈20.00087>𝛿.(5.2)The proof of Theorem 4.1 is complete.

5. Examples

The following two examples show the feasibility of the main results.

Example 5.1. Consider the following system:(𝑥(0),𝑦(0),𝑢1(0),𝑢2(0))𝑇=(0.31,0.26,0.02,0.08)𝑇One could easily see that there exist (0.18,0.18,0.01,0.1)𝑇, and 𝑥,𝑦 such that𝑢1,𝑢2Clearly, conditions (H1)–(H6) are satisfied. From Theorems 3.1 and 3.2, the system is permanent and globally attractive. Numeric simulation (Figure 1) strongly supports our results.

Example 5.2. Consider the following system:𝑥By simple computation, we can easily have𝑦Thus, condition (H7) is satisfied; from Theorem 4.1, it follows that 𝑛 Numeric simulation (Figure 2) strongly supports our result.

Acknowledgment

This work was supported by the Program for New Century Excellent Talents in Fujian Province University (0330-003383).